1.) For the arithmetic sequence {-10,-2,6,14,}, find:
a.) a recursive rule for the nth term
b.) an explicit rule for the nth term
2.) For the geometric sequence {512,256,128,64,...}, find:
a.) a recursive rule for the nth term
b.) an explicit rule for the nth term
3.) Determine whether each of the follo

1. Find the curl of the vector field F at the indicated point:
2. Evaluate the following line integral using the Fundamental theorem of line Integrals:
3. Use Green's Theorem to calculate the work done by the force F in moving a particle around the closed path C:
4. Find the area of the surface over the part of the

Find the mass and center of mass of the lamina bounded by the graphs of the given equation and of the specified density.
Use polar coordinates to evaluate where R is the region enclosed by .
Suppose the transformation T is defined by and . Find the Jacobian, , of T.
Please see the attached file for t

For what values of a in R (real numbers) is the function (1+x^2)^a in L^2
NOTE: let L^2 be like in Lebesgue integration where the set of all measurable functions that are square-integrable forms a Hilbert space, the so-called L2 space.
keywords: integration, integrates, integrals, integrating, double, triple, multiple, r

Please help with the following problems. Please provide step by step calculations and diagrams.
For the following problem, sketch the region and then find the volume of the solid where the base is the given region and which has the property that each cross-section perpendicular to the x-axis is a semicircle.
1) the regio

Determine where the function f(x) = x + [|x^2|] - [|x|] is continuous
I don't understand how to work this problem. Can someone show and explain how to solve this problem in detail? I've asked for help on this problem before but I still don't understand it.
The function is continuous everywhere but 0. I don't understan

1 Find the area of the part of the surface 2z = x^2 that lies directly above the triangle in the xy-plane with the vertices at (0,0),(1,0) and (1,1).
2 Find the volume of the region in the first octant that is bounded by the hyperbolic cylinders xy = 1, xy = 9, xz = 4, yz = 1, and yz = 16. Use the transformation u = xy, v =

Find the volume of the solid formed by revolving the region bounded by y=x^3, x=2, and y=1 about the y-axis.
keywords: integrals, integration, integrate, integrated, integrating, double, triple, multiple

Let F = (2x, 2y, 2x + 2z). Use Stokes' theorem to evaluate the integral of F around the curve consisting of the straight lines joining the points (1,0,1), (0,1,0) and (0,0,1). In particular, compute the unit normal vector and the curl of F as well as the value of the integral:

See attached file for full problem description.
Use Stokes' theorem to evaluate the surface integral of the curl:
where the vector field F(x,y,z) = -12yzi + 12xzj + 18(x^2+y^2)zk and S is the part of the paraboloid z = x^2 + y^2 that lies inside the cylinder x^2 + y^2 =1, oriented upward.

Suppose that a tank initially contains 2000 gal of water and the rate of change of its volume after the tank drains for t min is '(t)=(0.5)t)-30 (in gallons per minute). How much water does the tank contain after it has been draining for 25 minutes?
keywords: integration, integrates, integrals, integrating, double, triple, m

Let R be an integral domain with quotient field F and let
p (X) be a monic polynomial in R[X] : Assume that p (X) = a (X) b (X)
where a (X) and b (X) are monic polynomials in F [X] of smaller degree
than p (X) : Prove that if a (X)is not in R[X] then R is not a UFD(unique factorization domain). Deduce that Z[2sqrt2] is not a